ORBITAL FLIGHT

INTRODUCTION

Orbital flight around a planet is a balance between the centripetal force required to keep the spacecraft in the orbit and the available gravitational force of the planet at the specified orbital radius.

CIRCULAR MOTION

Newton's first law states that "a body remains in a state of rest or of uniform motion in a straight line unless acted upon by a force". In other words any deviation from rest or uniform motion requires a force and the magnitude of that force is specified by Newton's second law (F = dp/dt).

Now circular motion in not uniform motion in a straight line, so we need to calculate the acceleration of such motion.

With reference to the above diagram we see that although the magnitude of the velocity vector remains constant with a value v = f ω its direction changes. The vector of that change is dv. This vector points toward the centre of the circle and has a magnitude of v²/ r = r ω².

CIRCULAR ORBITS

The centripetal acceleration required for a circular orbit is v² / r. Thus the force required to maintain a circular orbit is m v² / r (Newton's second law).

Note that the orbital speed v_circ is a factor √2 less than the speed required to escape the planet (Escape Velocity ).

ELLIPTICAL ORBITS

If the injection speed of the satellite into an orbit distant r from the planet centre is not equal to the speed for a circular orbit an elliptical orbit will result.

If the injection velocity is too low the satellite will hit the planet within one orbit. If the injection speed equals the escape velocity the orbit will be a parabolic one. If the injection speed is greater than the escape velocity the orbit will be hyperbolic. Both these last two orbits will leave the planet entirely.

An ellipse is specified by two parameters. The semi-major axis specifies the overall size of the ellipse and the eccentricity e specifies the shape. The eccentricity of an ellipse can vary from 0 (a circle) to 1 (a parabola). An hyperbola has a value e>1. The period P of a satellite in orbit (travelling along the ellipse) around a planet which lies at one of the foci depends only on the size of the semi-major axis a. This period is given by the formula:

P² = ( 4 π² a³ ) / ( G M )

G is the Universal Gravitational constant, and
M is the mass of the planet

ORBIT ORIENTATION

It is not sufficient to specify the size and the shape of the orbit. We must also specify its orientation with respect to the planet (its equator and polar axis).

The important angles that specify the orbital orientation are;

• inclination i
• longitude of the ascending node Ω
• argument of periapsis ω

Angles are specified with reference to a plane or a direction. For planets, inclination is specified with respect to the equatorial plane, which is defined by the planets rotation. The argument of periapsis is also defined from this plane. The longitude of the ascending node, which is where the orbital plane intersects the equatorial plane, is defined from specified reference direction. This direction is called the first point in Aries (a constellation on the ecliptic), and is where the equatorial plane intersects the ecliptic plane. The ecliptic plane is the plane in which the Earth orbits the Sun and is inclined at 23.5 degrees to the equatorial plane.

The final specification is the time t at which the satellite passes periapsis. If the orbit is about the Earth the periapsis is called the perigee and if the orbit is about the Sun the periapsis is called the perihelion.

In total we then have six elements to specify the satellite orbit.

CONCLUSION

All the theory above assumes three things:

1. The planet is spherical with a uniform composition

2. The planet is the only gravitating mass to affect the satellite motion

3. The mass of the satellite is a lot less then the mass of the planet

If the satellite is man-made item 3 is usually assured. However, items 1 and 2 are usually an approximation, and sometimes may not be valid at all.

The Earth is not a perfect sphere and its small amount of oblateness means that a satellite's orbit will change within a few days. For high precision satellites like GPS new orbital elements must be issued every week.

For low Earth orbit, the gravity of the Moon and the Sun are usually not important. Atmospheric drag and the Earth's oblateness are the two more important factors. However, in higher orbits, such as geosynchronous orbit, the Sun and the Moon have a small effect. For instance, communication satellites must expend fuel to keep them close to their assigned orbital longitude - so that customers do not have to move their antenna to follow the satellite.

Around the Moon the situation is a lot worse. In low lunar orbit(<100 km) mass concentrations within the moon will rapidly change the orbit. However there are a few inclinations which are relatively stable.

Above 100 km altitude things get a lot worse. The gravitational effects of the Earth and the Sun then start to become significant, and orbits are continuously changing. There is one orbit that is reasonably stable and that is called a near linear halo orbit (NLHO) but that takes the satellite on a trip that varies from close to the Moon to a long distance away.

Australian Space Academy